LukeFarley November9,2017Partner:KarlensJoas PHYS113-05

ADynamicLookatourBungeeCord

Inthisexperiment,wesetouttomorerealisticallyinvestigatethepropertiesofourbungeecordbyconductingadynamicexperimentinwhichthecordwouldbehaveinamannersimilartothatofanactualbungeejump.Hence,wedesignedanexperimentinwhichwecouldusetheConservationofEnergytodetermineanew,dynamicbungeeconstant,whichwehypothesizedwouldbeslightlydifferentfromthestaticconstantweanalyzedinourpreviousexperiment.OurexperimentwasbuiltonthefollowingequationwhichfollowsfromtheConservationofEnergy,claimingthattheoriginalpotentialenergyduetogravityshouldbeequaltothefinalspringpotentialenergy.

𝑚𝑔ℎ =12𝑘(∆𝑥)

,

Here,histheverticaldistancebetweenthedroppointandtheheightofthemassatthebottomofthejump,and∆xisthebungeecord’sdisplacementfromequilibrium.Wedroppedamassattachedtothebungeecordandrecordedtheoriginalheighth,aswellasthecord’sdisplacement,∆x,tofindadynamicbungeeconstantk.Forthreedifferentequilibriumlengths,wedroppedfivedifferentmassesandcomputedtheaveragek-valueateachequilibriumlength.Wethenplottedthek-valuesagainsttheequilibriumlengths,andfoundamodelforktobek=47.232L2–48.402L+14.772+/-.0936N/m,wheretheuncertaintywascalculatedwithalinearregressionanalysis.However,weunderstandthatthismodelcannotbeaccurate,asitisquadratic,andthuspredicts thatkwilleventuallybegin to increaseas theequilibrium lengthbecomes longer.We believe our datamay have been skewed by the difficulty of collectingaccuratedatawithaslow-motioncameraandtape-measure.Figure1:DiagramofExperimentSet-up

Withamountclampedtothetable,wehungourbungeecordandtiedthehangingmasstoitwithasmallslipknot.Wealsohungatapemeasuredownfromthemountinordertoaccuratelymeasurethechangeinheightandthecord’sdisplacement.WeusedaniPhonecamera,specificallytheCoachMyVideoapplication,inordertocapturethepositionofthemassatthebottomofitsfall.Table1:Summaryofk-valuesbasedonequilibriumlength.Uncertaintywascalculatedbytakingtheaveragestandarderrorinalinearregressionanalysisforeachofourk-values.

EquilibriumLength(m)+/-.001m k(N/m)+/-.109115N/m

0.351 3.602323

0.448 2.567988

0.557 2.466327Chart1:TheplotofPEgrav(mgh)vs.1/2∆x^2forallthreeequilibriumlengthsL.

AsvisualizedinChart1,therelationshipbetweenthepotentialenergyduetogravityand½∆x2wasindeedfairlylinearforallthreeofourequilibriumlengths.Notice,however,thattheslopeofthelinesforequilibriumlengths.448mand.557marealmostidentical.Althoughwewouldexpecttoseethek-valuecontinuetodecreaseataconsiderablerateastheequilibriumlength

PEgrav =3.6023*1/2∆x2

PEgrav =2.568*1/2∆x2

PEgrav =2.4663*1/2∆x2

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0 0.2 0.4 0.6 0.8 1 1.2

PEgrav(J)

1/2∆x2 (m2)

PEgrav vs.1/2∆x2

L=.351m

L=.448m

L=.557m

Linear(L=.351m)

Linear(L=.448m)

Linear(L=.557m)

increases,thiswasnotthecaseforourdata,whichledtofurtherunexpectedresultsinattemptingtomodeltherelationshipbetweenkandequilibriumlength.Ourunexpectedresultsarelikelyduetohumanerrorinmeasuringdisplacementwiththeslow-motioncameraandtape-measure.Asinourpreviousstaticexperiment,weplottedtheseaveragek-valuesagainsttheirrespectiveequilibriumlengths.Chart2:PlotofkagainstL,whereListheequilibriumlength,foreachofouraveragek-values.

Afterfittingseveraldifferentmodelsonourdataandconductinglinearregressionanalyses,wefoundaquadraticequationtofitourdatabest,asseeninChart2,whichisdescribedbytheequationk=47.232L2–48.402L+14.772+/-.0936N/m.Othermodelshadhigherstandarderrors,between.25and.35N/m,anditwasalsovisuallyclearthattheydidnotfitthedataaswell.Intuitively,however,thisquadraticcannotbeanaccuratemodelfork,asthiswouldsuggestthatthek-valueswouldbegintoincreaseagainastheequilibriumlengthbecomeslonger,whichweknowshouldnothappen.Insummary,weconductedadynamicexperimentwithourbungeecordbydroppingamassattachedtothecordandrecordingthedistanceitfellaswellasthedistancethecordstretched.WeusedthisinformationalongwiththeCWEtheoreminordertofindanexpectedk-valueforourbungeecordatthreedifferentequilibriumlengths.Furthermore,inanefforttodevelopamodelforkgivenanyequilibriumlength,weplottedkagainstequilibriumlength.Sincewebelieveourquadraticmodeltobeinaccurateduetopotentialerrorindatacollectionwiththeslow-motionvideo,wewilllikelyconsiderthefindingsofourfellowclassmatesintheBungeeJournalwhenplanningourfinalbungeejump.Onmyhonor,Ihaveneithergivennorreceivedanyunacknowledgedaid.

k=47.232L2 - 48.402L+14.772

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k(N/m

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Eq(m)

kvs.L

kvs.Eq

Poly.(kvs.Eq)